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Using the probability distribution of a random vector , we seek
to evaluate the following probability:

Here, is a random vector, a deterministic
vector, the function known as limit state function
which enables the definition of the event

If we have the set of
independent samples of the random vector , we can
estimate as follows:

where
describes the indicator function equal to 1 if
and equal to 0 otherwise;
the idea here is in fact to estimate the required probability by the proportion
of cases, among the samples of , for which the event
occurs.

By the law of large numbers, we know that this estimation converges to the
required value as the sample size tends to infinity.

The Central Limit Theorem allows to build an asymptotic confidence interval
using the normal limit distribution as follows:

It launches the simulation and creates a SimulationResult,
structure containing all the results obtained after simulation.
It computes the probability of occurence of the given event by computing the
empirical mean of a sample of size at most outerSampling * blockSize,
this sample being built by blocks of size blockSize. It allows to use
efficiently the distribution of the computation as well as it allows to deal
with a sample size by a combination of blockSize and
outerSampling.

Number of terms in the probability simulation estimator grouped together.
It is set by default to 1.

Notes

For Monte Carlo, LHS and Importance Sampling methods, this allows to save space
while allowing multithreading, when available we recommend
to use the number of available CPUs; for the Directional Sampling, we recommend
to set it to 1.